 Генеральный момент, который partly motivated by your mathematical issues, partly by application, is that probability theory, probability is not a number assigned to something, but rather I say what function, though it's not exactly a function from category, which is not quite a category of complicated object to something relatively simple. And we have events in the world and we assign to them numbers and numbers make a category of positive numbers. And that's with certain properties and that is probability theory. But in many instances application I won't discuss, like in biology or linguistics, this first set of events is structured and rather take away very differently from physics. It's highly non-homogeneity, so it's organized in a less elegant way, but still quite powerfully. And the range where probability takes place, neither it is a number, not a number. It's something which is not quite clear what. And the parallel which you may draw from your mathematics is homology theory. When you start with category of topological spaces, you complicated things. But what you assign to them are a billion groups or even better vector spaces of a field. And this is something you feel much more comfortable with. They measure in a way, they serve for measuring your objects. And one of the ways to create such theories out of what you know is linearization. And one of these standard ways to generalize theories is linearizing them. And just one of them, just the point I was making, entropy and some aspects of the entropy are linearizable. And the simplest instance of that is what I was stating was the Lumisuitini theorem. Isoperemetric inequality is quite respectable because it has lots of consequences. Sets in a form which we want to generalize as follows. We have a product of sets, of finite sets, which may be infinite, but... And we have a subset there, which is y. And we project it to different products. We project, you can see the y. So we write this y sub j. It will be a projection of y. And this is the product of all i and j. So there is a projection from x. I'm sorry. Ah, yeah, that's correct. So x projects all these x sub i, x and j. And y projects. And then the cardinality of this set is related by some inequality, namely the cardinality. So now, for the case of Lumisuitini theorem, we only project... So we drop only one index at a time. So we project here, we project here. And so there are projections of the coordinates hyperplane on the top dimensional. So we speak. So there are. I know them, and so you can call them maybe x i drop, meaning this index being dropped. And accordingly we have this y projection i being dropped. And their cardinality, their product. Product of those cardinalities. And it's better to normalize them to the power 1 over n. Great or equal to the cardinality of y. If my normalization is correct. If my cardination is not correct, I guess. I think this is the correct power. Yes, that's correct. So you're taking these powers, these projections. So each of them projects on all products, but one. So one is missing. So a total number of them will be equal to say n is cardinality of i. So I'll put here n minus 1. Total number of them is n. But, for example, dimension 3, the first example to have in mind, you take powers 1 half. So we have subset in Euclidean space. You project it on three coordinate planes. Then product of areas in power is 1 half. Bounds the volume of the set. Which is discreet and continuous. One goes to another coming in a trivial way. And then this has kind of, I'm using generalization. So this is a Lumi-Suite in inequality. Which is actually become more transparent if you generalize it. And this generalization is called Sherwood Lemma. And it is as follows. So we can see the function alpha on J's. So this is a function which takes value in real numbers. Poetry of real numbers for all J and i. So we can see that the following function, which is called a partition of unity, if it's not mistaken. And now, because I have a numeric tag, it's alpha and take it here, chi of J equals 1. So you replace EJ by character, you indicate a function of this set. Function 0, 1. And you take this sum. So in other words, when you consider this set J, and here how many times you cover a particular point, and take this way it's A, alphas, sum must be 1. So it's called partition of unity. In particular, our choice here, these all exponents were equal, and that's exactly makes partition of unity. So in dimension 3, we have 3 points. So we have 3 sets. And each of them covered twice. So we have to take 1.5 to have partition of unity. And then the corresponding inequality says that if you take product of cardinality of Y sub J to the power alpha of J, this will be greater or equal than cardinality of Y. And this is a share. Let me again, I don't know exactly the history, when, how, and where. It was exactly stated in each context. And it's kind of people love it in information theory and formulated in a form, in comprehensible for mathematicians. But it is a formal consequence of strong supianitivity of entropy. So first you can strengthen this instead of taking this projection, you take push forward measures, which are entropies. So you replace them by smaller entities, each of them. And still inequality holds true, because here, of course, because subsets are the same as its own entropy. You project it, and it still remains true. And for this entropies, so if you just apply the strong supianitivity to all kind of covering by pairs of sets, issue them. To issue this block you apply and use what's called inclusion exclusion principle. You start counting and you get this inequality. It's still kind of nice consequence of that, but still kind of completely formal. It's just linear A plus B mathematics. However, it's nice, nice statement. So what I'm using, it has linearization, in which I want to formulate topologically, which is kind of easier to formulate, because it becomes much more close to what we have. Namely, now let X, I's be topological spaces. That kind of simple space, maybe Euclidean space, but still they have topology like circles. Or whatever. And you can see their product as before. And do everything we were doing before. Now we have this subset. They open or close the subset Y in X, and you project it to all this product, call them XJ, product only by I from J. And then the same inequality holds if this understood as a rank of homologes, of these spaces with coefficient in some field. And in case, of course, of discrete sets, it will be the same. But in general, something significantly more general, much more possibilities, of course. And this has nothing to do at this stage, has little to do with topology. It's purely in algebra. And the only thing which is relevant is that homology of such a product with coefficient in the field is just tensor product of corresponding vector spaces of each representing homology of each of them. And that's all we should use. And you see that it's a kind of ugly thing to do, right? You have a space and you project set. And projecting image of a set of constant topology is definitely something wrong. So you would say, Hi, I would prefer to speak about rank of corresponding homomorphism. And then it becomes not true. Because may cancelation, even in zero-dimensional case, homology on the projection may go to zero. However, it doesn't go to zero if there is some positivity. And so in algebraic geometry, for example, it's still being preserved if you work with a, not with homology, which our group is a ring of cycles when there is built-in positivity. So positivity is kind of crucial. But they don't know if you can make entropic version of that. You see, this algebraic geometric version is not, it has no entropic counterpart. It is no apparent entropic. There is no strong subentitivity in this language. It just fails to be true. There are counting examples. But there are so special counting examples that there might be still probably right theorem. But this theorem is true. It's a starting point what we want to understand. So what we want to understand, what we want to say, homology in some respects, when you have a space X and you have open subsets, then homology of this with some coefficient, so co-homology, with fixed coefficient field. If I'm speaking with co-homology of U, in many respects behaves as measure. И, of course, homology of theory has built-in additivity, which is like exact sequences or mitotic formulas, which make it looking like a measure. Which, in fact, I don't know how to fully use. So I will be using very little part of that. But the point is that if you have all this property in homology and can say, well, we know everything with respect to sequences at that time, we know everything. But from a certain point we know zero. Because this is like in measure theory we know everything about real numbers. But it doesn't mean you know anything about measures. There's only starting point from which you can make formulating problem, but not proving anything. So let me explain whatever you have in mind. The point essential in probability theory when you do interesting problem, you always have infinite processes. Infinity is organized in a certain way. It's not just infinity. It's structured in a certain way very often by action of a group. If an infinite group and something happens homogeneously statistically along the group then you want to understand what happens. And here it's kind of a similar situation. You want to consider a similar situation and just look at very simple examples. So don't just see that something. I gave one example of this where topology was not quite simple, continuing with this example. Look at very simple. So we want to consider homogeneity situation when you have infinite families of topological spaces and sometimes maybe even infinite dimensional spaces. And we want to understand this asymptotic behavior of the topology when dimension goes to infinity. So the simplest example would be have a space x and you take a clutch power n and n goes to infinity. Let's look already at this very simple example. We want to understand how homology behaves. And again, my homology or homology always is a fixed coefficient field. You take this and you know that this will be tensorial power of this. And particularly upon correct polynomial of this space meaning polynomial which coefficients are better numbers. Right bi meaning ranks of homology in some dimension become power polynomial. And the question is what is asymptotic behavior of coefficients? And of course even for binomial it is these numbers. The asymptotic is exactly what is given by Boltzmann rate formula. So I would think that this formula is asymptotic behavior given in terms of entropy. So in fact there is some specific formula saying how they go. Of course traditional proof you just apply sterling formula which is absurd because the sterling formula about n factorial there are two terms n to the n and then there is exponential term. Something like e to the n. And but this cancels this major term asymptotic cancels in there. Right? And so only this remains so it's clearly that sterling formula completely wrong way to do that. Which is on the other hand of course you know the formula inclined to do that but it's exactly what I'm saying should be prohibited of doing. Right? Because it's completely distort the picture and on the other hand this is if you think about that in terms of entropies it's clearly is the number of states of certain system and if you think about this a little bit you see it's actually entropy as defined by Boltzmann in terms of his alternative definition. This is how Boltzmann proved it but again you could think it was known too early but they look at that on the internet there are some articles by earlier about that but he doesn't prove that for different binomial coefficients for certain kind of polynomials but it doesn't seem he was concerned with these asymptotes. So I'm curious who first found asymptotes with binomial coefficients because this is of course the core of large deviation theory. All large deviation theory is a liberation of that. He just in a way wants to know that the rest you have to know what to do. So this is the key key in one of the most fundamental issue in probability asymptotics of binomial coefficients. And for general polynomials slightly tricky but still not hard polynomial one variable so you know how they are formed you can write these formulas and so even asymptotes of this coefficient not completely trivial slightly probabilistic in this nature. However the issue which appear usually probability have discrete sets and their powers and everything refer to zero dimensional homology he automatically have the pleasure dealing with all dimensions. So the thing become graded and there are all dimensions the picture becomes much more interesting and so let's see what we can say about it. So so our object is x to the n n very large going to infinity but I prefer to think about that it's not infinite now but infinite to large but still integer and we want to measure sets by data apology and so what would be the good measure just their own topology is not quite right yeah because kind of physically you think for the basic example for us for this moment will be n torus or I better to say we torus to the power set because we can imagine have set on issue have a molecule where the state of set of states of this molecule the circle all together we have the kind of very large piece of a crystal and each of this imagining this being non-interaction which is of course very rare in physics so if these would never consider that it's kind of trivial they don't interact but still even that indistinct and interaction is kind of another matter and topology instead of saying the measure of probability of event we want to know how many motions a particular state may have right so we constrain we constrain our picture by some way we observe and for us it may be some open set inside of this torus to the power n or rather power i this is and it's how much motion we will see how particles can move here of course motion rather simple they don't interact the motion don't interact it's kind of trivial case but as we see there are many are not open questions here and there are some results and so what is the correct way of measuring right so and the way of measuring seems to me the best because there are two competing kind of way of measurements which set a large and which are small so if we have just in the euclidean space such space and we take kindly little holes of course we have lots of topology but they can fake yeah of course if you erase them they are not monotone yeah we erase these points which we removed so add them back set become larger homology become smaller so this you cannot use you need some monotonicity and of course so what you do you consider instead of that first definition it is a rank of homomorphism homology homomorphism of the whole x to y so rank of homomorphism now of course it's better because monotone for sets so you have positive function which has this monotonicity but it also has some this is already not so bad and we shall it may be used but the better measurement in fact is somewhere different and which has more kind of measure kind of behavior and t is what we do we take our set u and we take rank of I'm sorry yeah it's a rank of the kernel of the homomorphism from x it was y I was calling the mu now the kind of open sets so to mean from x to complement to u on the homology level in another way to say it you look so we have the subset in the big space and you ask which homology of the big space is presentable by core chains with support in this set and which corresponds again by elementary typology to the kernel of the method on the complement so it's a rank of homology of the ambient space representable by core chains which vanish outside of this set and see this behaves in many respects like like a measure and it has some nice sub additivity property with respect to cup product now homology what is nice something you can add so the object so you assign now the true kind of way to say it you don't assign to set a number but you assign to each set exactly this kernel so it's ideal, great idea in the homology algebra you take homomorphism from ambient space x to the homology of the complement so your measure takes values from certain numbers but in graded ideals of this homology algebra and with respect to them you can add ideals you can multiply ideals and the basic formula which is systematically used is that if you take so this you call a measure call this I put somewhere some stars so it's not usually a measure of you right and this measure of I hope I'm not confusing it it has some you see what I'm saying is a reinterpretation of the Listernich-Niedlmann theorem that if I have so one vanishes here and another vanishes here then the cup product vanishes on the union so we have two homology classes one vanishes somewhere another vanishes somewhere else we multiply them and vanishes in the product of course but this is kind of Listernich-Niedlmann theorem which has lots of applications kind of obvious ones being said in these terms but it's not at all consequence of this and obvious and this measure always forgetting that to put any quality in the right direction because otherwise I make it wrong this is so good what I have no ability to remember formulas so simple ones always put the sign wrong so it is said that mu1 U2 is contained so we take intersection of two sets this is contained which is exactly unable to remember to put the right sign it contains so this remember, this ideals this cup product it's not union, it's cup product there are some IDT which also we have just joint sets this thing is additive which is kind of trivial it's not that it's significant but this novelty extra formula I'm sorry it's for kohomology just remember it's for kohomology absolutely there is actually more to that I may not explain it that this not only this kind of idea it's graded algebra it's not only ideal but also invariant if you work mod 2 mod zp it's invariant so it's more structured and that is again significant role especially in the work by Ladi Guth now so what is what we know about that in the simplest example so this again this is just background but I'm using formula this is the following if you look and this of course you can do it restricting this you put here instead of that you can put some number which means it only refers to kohomology but of course for product you know what it will be so you can replace star by specific index and so there is the following theorem for the end torus so if you have two sets u1 u2 and then you have to write it down because otherwise I do it wrong there is some inequality for that which is also in a way similar to isoparametric inequality and so you take this means the ranks and this less or equal to what is the product of the numbers times these are basic numbers of the torus themselves so they are binomial coefficients and here there is a coefficient c by which you multiply and this c equals something but I write it but then I have to explain it and this constant c equals i1 times i2 almost there so n is our torus these numbers this is a constant c but this inequality holds if the states are disjoint so we have two disjoint sets then they cannot contain both to much homology and of course you say everything about homology you have all the sequences still you have to prove it and the point is the way you prove it so how I was proving that so I just reduced this to something combinatorial and then I was trying to find it in the literature some combinatorial statement and I was imagining all possible words but of course I could never guess that this was called corresponding combinatorial theorem was called Matsumoto Togishugi theorem I mean this is a horrible way how people call their theorem after the names who know these names and I actually I spoke to somebody in combinatorics and he directed me to the right domain of mathematics it's called extremal set theory and you look at textbooks and then you find lots of inequalities and so this particular inequality if your subsets are just union of tori which are coordinate tori which are product of circles so then you can express it in terms of sets and then so using this property you reduce it to some property of the exterior algebra and then going from exterior algebra to sets well that's again little simple argument but the point is again this is very much like in the case of the inequality of Shira Lemma again using these names which is of course and otherwise you have to invent for any purpose try to invent actually some names of the theorem so they will have some meaning and then you will be remembered because that's name being attached except for no Pythagorean theorem is okay we can remember the name but if it's come to somebody we proven yesterday if it's really good theorem only exceptionally name is okay then kind of linearizes and then become topological in terms of linear algebra in terms of exterior algebra I just don't remember how to formulate it it's because of you and this a pretty certain theorem is true not only for torus but for all kind of spaces but I'm not certain because the corresponding I think the rest combinatorial theorem have been proven you need something more elaborate combinatorial theorem generally it's unclear what are kind of full set of such inequalities so you have configuration of sets so this is a problem I can formulate as follows so we can see that n goes to infinity and you can see the configuration of sets in this X of n they are indexed by the same always the same number 2 of course not the same as it was before just here we have 2 subsets now some configuration of subsets and what you know of them you know who intersect whom and all data which you have all which is more important who is disjoint from whom and then the question is what are possible measures in my sense of these sets so how much topology of the ambience they can absorb asymptotically n goes to infinity and this is not so clear I mean it reduces to some question in polylinealgebra and sometimes you can reduce at the combinatorial question sometimes not sometimes the corresponding reduction look questionable and then you arrive at some combinatorial problem in this extremist set theory which usually are unsolved right in this generality I don't think it has been done but that tells you something about the kind of problems we face here ok now so what what we can do next so this kind of a warming up but this warming up where I can prove something from some moment on you can prove much less so you can make some definition of formulaic questions so one of course immediate issue which arises as a statistical mechanics one way you say aha you are a big system, a limit of finite systems you want to describe everything for finite systems and go to the limit and alternatively you want to work with infinite system to start with and in statistical mechanics if you just abstractly infinitely many objects you cannot say anything but if you have a lattice namely you have some group gamma acting on there then everything must be done equivariantly with respect to gamma for example topologically you have this infinite dimensional space so it will be not just torus to power infinity which is unclear what meaning it has but it will be torus to the power of this group so it will be that of function from group to this torus which is acting upon the gamma and you want to make this equivariant topology there and the essential part of this topology is multiplicative structure additively it's not that interesting but multiplicatively it's kind of the whole kind of essence of that everything what I said depends on multiplicative structure of course this relatively simple space and well you can you can kind of much of them derived from from product but still there are issues which are not completely obvious here however on the next level if you want imitate what happens in physics particles when your factors interact you have to introduce interactions and then you follow the patterns of what is done in in topological dynamics symbolically dynamics so in symbolic dynamics in topological dynamics we start it will be not a torus but a finite set you take it to the power gamma and here gamma x so it's infinite power finite set and this power gamma x and that out of this you can produce more interesting dynamical systems namely what may be called Markov hyperbolic systems which includes classical hyperbolic system like anosov systems this is done in two steps and these steps can be imitated in this more geometric topological setting the sets as follows first we consider these are called sometimes shifts they are used between those shifts this is kind of Bernoulli's shift particularly when gamma is integers you have this find too many symbols infinite words the symbols you move the basis and the infinite words move first we consider subsystem which is called subsystem of finite type it is defined as follows for any group any countable group gamma they define as follows you take some subset in gamma finite subset in this context is a function with values in a set here it is finite set here is something like circles something more interesting and you say looking at this one particular subset which configuration you are being allowed so you take only subset of this configuration so you can say you add finitely many equations if they are discrete of course equation makes no sense if we are continuous you can say how finite variables finitely many variables which come there and they satisfy some equation so this is a finite dimensional description or finite here and then you say now we can see that only those function on the big on the whole group which satisfy this equation whenever you put this domain you can shift it around the group and it requires the same condition being satisfied everywhere for example we have everywhere this equal every of these two this is a harmonic condition like harmonic functions appear this way the same condition everywhere whichever and these are called finite type subsets that's a much of the symbolic dynamics is about that here you can do the same of course for the torus I don't know what it may be but nature of good candidates when you expect something regularly happening of course algebraic varieties replace the torus by projective space projective space over complex numbers and when you are restricted here so you have a projective space so here is a product of projective spaces and there you take some algebraic subvariety and this will be allowable and then you move it around and then you have infinite infinite dimensional algebraic variety and the simplest instance of that when you run over integers and then of course you can truncate it at every moment and you have a finite dimensional variety or you can do it periodically in other approximation and you have this again finite dimensional variety so given one single equation you have this sequence of varieties and the question is actually what are they at apology and this unclear I mean I was asking many people in this way exam to say oh well we have all the theorems on the beginning the theorems only can like a plus b equals c you have to make some hard work to understand how they behave you know in this way exam the apology is very stable you move a little bit equation in some sense the apology doesn't change so it's really a kind of problem there must be canonical answer of your varieties however they may behave in a rather complicated way and if it comes for a more complicated group gamma it may be enormously complicated but for a cyclic group probably it's solvable but if you start thinking about that you see you run into rather into a complicated mess you know everything for every finite stage you have full picture but the picture becomes more and more elaborate very fast so but this I think interesting question however maybe you will never get sharp answer and by the way another point the way you can approach it of course if you work on a finite field then if you reduce a prime p you can see the prime p then you look at the points of a finite field you come back to the Markovian chain so you come to the classical dynamics so it has got interesting interaction with dynamics and homology and so so this ranks of homologists in this space are closely related to entropies in the usual sense in a very precise way however the picture is far from being clear however it is not and then maybe just to continue to finish to explain that this is not the end of the story especially here because here there is very nice next step because you can consider quotient of this so this is a kind of traditional this is less traditional you can see the quotient of this fine type systems and quotient must be understood and again in this category namely so the quotient space of any space obtained by can be introduced by by means of an equivalence relation an equivalence relation is a subset in the product of that by itself so this equivalence relation you take two of them and you take subset here and you say whom you have to identify and you only work in the category of system of finite types the way I describe them so describable finite terms so this subset, this equivalence relation are must be also a finite type on the other hand it must be equivalence relation it must have the quotient and these are two properties which fights each other it's very hard to have both of them if you want to construct and this of course must be in this everything automatically will be of course equivalent of the action of the group the way this I described so you want to produce equivalence relation of finite type this is very difficult to have it's very hard to verify it's like a metric if you have function in two variables it's very difficult to say it's a metric or not it's really the most impossible it must be defined and here it's quite difficult and much of difficulty of dynamics come from that problem so what are these systems they kind of hyperbolic system in this generalized sense but here it's unclear what happens if you do it on this level of algebraic varieties it's unclear if it's significant but with them it may still remain in the same category but that's not the main issue what I want to explain because there is much more interesting for class systems where you can apply this approach and they are physically quite attractive so these are all close kind of non-interacting particles so everything what you can do there even if they don't interact or interact in a kind of not very significant way hyperbolic system corresponds to that but so what you want to consider what I call is parametric packing problem which has much more juice to it not that it's more the previous one not difficult enough but there you have really difficulty just when you here there is more possibilities more difficult but probably easier and that starts with the classical pattern so I have a bunch of balls in the euclidean space and you want to know how much you can put them with a given density and it's a it's unknown what kind of solution you should have where the only where it's solved in a way only in dimension 2 and 3 2 classical going back to the genre the optimal packing is hexagonal one so it has additional chain you know exactly what it is and you know what it is in high dimensions you do this you can put one layer or another this was conjectured by attribute to the captain it was recently solved I forgot the name of the guy who solved it and it's computer-aided proof accepted and what computer-aided proofs say in a way I don't quite understand objection against computer-aided proofs or rather any proof obtained by computation which you cannot do in your head is like that some of you make computation with the paper I mean how you trust the paper why should you trust the paper it's a paper I mean it's physical mean not your brain I don't see a big difference you have argument which don't requires something you cannot have in your head and long linearized things arguments are not good for your head of course you may not accept it but anyway his solution is not that simple it's not periodic they don't have to be periodic they describe all of them and when you go to next dimension you don't know what happens most likely the extreme thing and what they are and how they look lots of interesting questions but they are not what we want we want to look at a course question but which has more and more substance to it in my view is when density is far from being maximum you consider both spread with some density which far from being maximum yes you have you know the density so both are distributed such that in a bigger ball they are proportion given to you and you want to know topology of this space you can see all possible positions and you want to know topology in a sense just I was speaking about that what is asymptotic behavior of homology but even formulating it is not so obvious on one hand but on the other hand I can say already now about that and the point is that topology here is by far more elaborate than already background topology language in which you want to have an answer will be much more elaborate than we had before it's not just infinite tensorial power of some vector space it's something more complicated so let's make a little break and we come to this okay so that's an issue so we want to understand this and as I said even formulating it is not totally obvious and it has kind of obvious kind of physical interpretation they have it's moving balls and you want to know how much how complicated the motion can be and they can go one you know something like that and this orbit is in that one and you want to have a picture of all this complexity and of course this homological setting is a possibility to express it but not the only one there are several ways how we can formulate it and here I emphasize this homological formulation which is not the only one possible so what is so and you know people love energy and so let's look from the point of view of energy and just try to make everything about energy also formulated in appropriate language and then we come back to this problem just to formulate it in more general terms typically what you want to understand something the only way to understand it is when you focus on this and just and on the other hand you just go away and experience shows if you focus on something you can never understand it you know what is substance what is that you just know it but if you go away I mean you come back and you understand it or you understand it it creates you are stupid to start with so you have to go away notation maybe not consistent will be x and you have a space and you have functions with real values and this is for us energy so how this is treated in statistical mechanics in very simplistic terms this x is a probability space a measure space something like they are all more related the same something like square and you have a function to real numbers and this has this density distribution usually spaces depend upon parameter they are bigger and bigger chunks of matter so you have a sequence of these distributions but there are measures after all measures on the line and you won't understand these measures and for measures on the line or if you have several observables not one energy several observables you have measures on the eutidian space the first what you do you look at this entropy what is a asymptotic decay of this one usually they concentrate one point and then spread becomes very small otherwise and then you do all this formalism of mechanics and of course the point is in my understanding all this mathematics is kind of trivial but remarkably it's very non-trivial for practical physicists or chemists or biologists from that you know what to measure how to relate these kind of mathematics to quantities physically can measure etc and if you throw this away from mathematical stage mechanics you become complete boy in my view you completely have pointless exercises and computations the essence is that you it's related in non-trivial non-digorous way to experiments but we do mathematics however unfortunately and so but what we do now and typically if you have this this space with x to the n very often is a power of fixed space identical particles many of them it's not will be not this thing it will be thing divided by symmetry group and mathematically inevitable you have to divide by this group because you cannot make asymptotic with factorials but you have binomial coefficients as I said factorial can solve you have exponential functions and exponentials are just where you can make all this kind of formalisms of large deviations which is now called or which used to be or just gives kind of gives formalism of stage mechanics and now but if you have do topology so now x is not measure space not probability space but topological space and we are concerned with topology of this space and also if even it's a product we have for example in the case of balls everything determined by these points so it will just power space if there are n points and now the symmetry you have a symmetry group acting but division is quite different thing when we divide topology we put two lines and because we concerned with homology and homotopy you don't take set theoretic quotient which is a stupid thing to do we take homotopy theoretic quotient so in a way that you remember what this symmetry do to your space if you divide it by like that topologically become rather boring and not always boring but relatively at least it's not what we want because here in particular when we have these balls these points are disjoint so you are concerned maybe not so much with the space itself but this minus all is diagonal this means all diagonal and here division by the symmetry by permutation group is the same but you put this second line or not because the action of the group has no fixed point there and this is enormously complicated space topologically because this group is topologically very complicated so group in again classical physics power of the group come via quantum mechanics because x linearly has representation so this representation corresponds to more interesting groups in this response sometimes I don't want to say which I don't understand but homologically this also the homology of this group in particular when it acts on a space on a relatively simple space like that this group acts the quotient space is quite complicated quite elaborate homology so I can say what they are and they kind of known and textbooks about that but they must admit I never quite read it so I just couldn't extract from that information which I would like to know for example what asymptotics of these Bayesian numbers n goes to infinity in different range that is elaborate theorem saying what they are but they kind of don't mind at least the books not written what they are in simple terms and you have to work on this they go exponentially or non-exponentially however there are part of homology which you can understand but anyway what to do with this how to use the full information in that but now coming back to most theory so what is the setting of the most theory I prefer it to be energy and what about what people do in physics what will be spectrum of energy and the basic example of because this what eventually will come up also in this packing problem it will be question about the spectrum but understood in topological terms and so what is the spectrum so the basic example which everybody knows we have Laplace opérée and we know its spectrum and this spectrum can be seen in terms of the quadratic form associated with Laplace opérée and this is energy square of the differential and this is quadratic form so I always prefer to work with quadratic form rather than with operators because their topology becomes more visible and so what you have you have this this energy is integral of this you have some manifold X and the spectrum you restrict this function so it is a function on F you restrict this to the sphere in the Hilbert space so you have a function on the sphere again sphere has little topology means algebraic topology meaning homology is none in the infinite dimensional space it is symmetric therefore it descends to the projective space and this is more interesting because topology is just generated it is infinite power series one term in each dimension so it is polynomial ring and one generator so it is polynomial ring so there is one generator in every dimension and the spectrum is quite simple to understand much easier than algebraic spectrum because it uses much less input so you have a space you have a function and you look at the level imagine there is a bottom and then it moves and then space becomes bigger and bigger but you now measure it in homological term instead of taking mass of that which you don't do anyway you take this what I call this mu star f-1 of say minus infinity so there is this function and you take it which implicitly is kartoff kartoff meaning I replace kartoff replaced by number because this was ideal but you take a number rank of this and so it changes and the moment it changes are the moments with the spectral points classical variation principle relay I guess that spectral points critical points of this function energy function restricted to this sphere because symmetric it's projective space and it's a critical point not just something vanishes it's not local phenomena it's global so you absorb new cycle at every moment so in projective space it is circled and there is projective plane so there are cycles of high and high dimension and when you as your level grow you engulf one by one all of them and the moment the first moment you engulf the cycle corresponding to this homology or cohomology it is I's eigenvalue so the eigenvalue enumerated by these elements of your cohomology specifically there are this ideal the kernel of homomorphism actually to the complement because our measure associated with this actual ideal so your spectrum index in this case are indexed by homogenous ideal in this polynomial ring which happens to be the same as integers and then come this very beautiful point which is partly motivation for what I want to say is that there are cases when this algebra is more complicated and one of them for this packing problem but there is much easier case when this algebra is still complicated which has been studied by Lighty-Guth and where spectrum where this algebra is polynomial algebra in too many variables and moreover it is acted upon by it is generated by a single element but not as an algebra but as a module of a standard algebra so cohomology mod 2 acted also by standard algebra and there is a simply a kind of algebra there the one which is free is generated by a single element but it's as an algebra it is rather elaborate it has infinite main generators as an algebra an example that's quite simple the first example what happens three-dimensional ball and you can see the space of curves in this ball and this space properly understood the space of curves in this ball topology of this space is exactly what I said it's cohomology has one generator in dimension keep forgetting 2 1 generator of this ring they are polynomial but all of these are powers, standard powers basic generator 2-dimensional generator there 2-dimensional but how you understand curves first curves under the cycle either closed curve or some sequence may have ends on the boundary and they are just simple curves but you allow the kind of deformation you allow this picture you go from this to this so they merge and they diverge and you can see that this is continuous deformation and that's essentially the rule you have to say it slightly more carefully but you allow a little bit more deformations and this is a in technical terms the space of z2 cycles is coefficient of this group the relative cycle they are boundary on the boundary and this space has cohomology and this kind of behavior and the function which is of the interest here is the length of this curve and so in the spectrum of this energy you know to understand it you have to enumerate this eigenvalues by this kind of rather tricky thing it's ideals and this algebra and nothing you can do but you like it or not it's very good all this structure is there naively you can say all this relevance is only something if you use something simple it does not so it's the whole structure this algebra inside of this geometry or spectral behavior of this functional and this one of the easiest kind of geometrically thing you can imagine right first kind of non-linear first non-linear case and it's become elaborate this is our general setting is now we have this space we have this function and we want to understand how the spectrum how these levels change when you change them and what happens what happens in the case of the packing problem that the function you can consider your space will first reduce consider the case of finite systems and the ball which is by the way not you cannot say higher than a go to the limit it's tricky but it's simplification just to not to go to the next level of difficulties so your space is just configuration space of points so what you work with you have say a ball so it will compact manifold our ii is a finite set minus diagonal and the basic structure already this has rather complicated topology if axis even axis is rather simple even x like a ball this has quite a bit topology but moreover it's acted upon by permutation group automorphism with indices if I throw an index freely so it's a very elaborate object it has very elaborate topology and the energy which you use and which you want to understand is it will be just one minus minimal distance between points so in this way you can encode fact the points are far away so being balls being disjoint encoded by the size of this function so the closer they come the higher the energy so they don't want to come together and in this way you can describe so the space of of this moving balls is just sub level of this function and so you want to know so what are spectral moments how the spectrum behave for this or that palette and then so let me and the point is it was very good and it was in simple case was known earlier that there is a non-trivial interaction between these two problems so one you have balls moving here in this ball on the other hand you have these curves in there in the space of curves and between the two spaces there is kind of a coupling there is kind of duality between these two spaces in fact this is much easier space this is all the standard house compared to complexity of this space however some part of this space is reflected here in this part which more or less possibly the picture is adequate for that part of that and so let me explain let me explain this and I want to gives kind of a the spirit of that is as follows how we can think about what I'm going to say about the space of balls instead of thinking of this original packing problem like you've given a space like you've given a space you want to understand how this balls live there and how they move and what is the packing you can say can you think of the whole manifold it's just container for packings and forget all other structures so even the Hermione manifold for every size of ball every number of balls you have the space and they move and then the spaces they are kind of not separate they organize in a variety of ways because you move size of the balls then if you think if you know a little bit some jargon if one of them is a ball playground space they called an operator there is operation simple mining because if you put little balls here inside they will fancy-fancy operator so you see quite all these packing spaces well organized and they are purely topological they depend on some real parameters but they are the homotopy types of the spaces and they organize in some way so it's a elaborate collection of homotopy class of space and error between them what they tell you about the manifold so what is information they tell you about the manifold and so just to demonstrate this particular instance of what is here consider the case whether it is especially simple when you have a manifold I say this example just because it is most transparent of compact this being compact manifold of negative curvature and then the point is that you can essentially construct geometry from essential geometry probably all but essential from knowing these spaces namely you can determine the length of all closed geodesics and this exactly the spirit what I said this kind of curves there is a coupling between the two and so this is what I want to explain how things are coupled and it is simplest example but in general it is somewhat more elaborate than the end of the story so we can see the manifold looking like that if you do not say it may be surface here is a closed geodesic and it opens up so it is negative curvature it has this narrow neck and then opens up and we want to understand what is the length of this geodesic in terms of disks moving inside and the point is follows if you have K disks and now symmetry will be not involved yes it is very primitive kind of description but already not so bad so they can go from this side to this side and quite freely but then they can go kind of back and forth and so of course topology here must be understood kind of relative to infinity but if total sum of diameter of these disks less than this length then you go back and forth independently so if you close it up somewhere then inside we have this product if there are K of them then this homology class which is torus this homology class of the fundamental class is in the space of this in the space of these moving balls because when they meet together here they don't this is kind of one from another but if the sum of diameters become greater so that to many of them this cycle is not there so if you precisely know the length of this geodesic knowing homotopes, homologists even this relative homologist of the space of disks moving in there and that's true for all dimensions so what is essential the space outside of this narrow neck become wide and so this is kind of the spirit of this interaction that they kind of make in general there will be not this narrow neck but this kind of virtual inside some entities which when corresponding to corresponding to eigenvalues of this space so here all closed geodesic will continue to be part of the spectrum of this energy function right so it's very elaborate space which actually somewhat in the spirit of you know this trace formula which is very different from that of course typological so it's kind of typological version here of the celbic trace formula when spectrum will a plus impirate on one hand and distribution of periodic orbits are related in this way so this is a formula and and so this is a quite quite quite a general principle so there are two sides to that when you want to use it on one hand and this was done by Larry Goose he constructed this most major part of his work quite elaborate movements of balls of small balls how they move inside of another representing interesting cycles inside of the ball because here it's typology coming not symmetric typology but symmetric typology though not for the whole symmetric group but only for the kind of maximum celove 2 subgroup only concerning convolutions how much they determined by this celove subgroup there and on one hand on that hand and on the sides on this part says there are complicated motion of balls representing you can put relatively many balls into the container and they still can move with sufficient degree of freedom and then there is opposite type of estimates you can say how if you have too many you cannot move too much some class is not representable and for that again you can use this kind of biology if there is obstruction there is narrow cycle inside narrow circle but in general it will be not individual circle it may be moving circle which have the same effect here it was fixed circle moving balls but in full generality you have moving curves and moving balls they interact so in a way that these curves intersect ball and move within balls and it give you inequality between two spectra so one functional one spectrum they described length on curves on one hand and distance between inverse distance between these points in the space on the other hand so there is this coupling between the related two spaces which is probably not faithful I mean it is still far from faithful probably for cycles asymptotically correct information about cycles but hardly give you the full information about the balls which is by far more by far more elaborate space but then of course it's only beginning because even if you know all that you have to now to go to infinite systems and then it's also clear how to even formulate it what is the density, how many balls etc etc there are many typologists and something which you can say but not much so I want to stop here because I want to maybe to answer questions because my last lecture and I can elaborate this of that point but from this moment even little I can say become rather technical it's not it cannot be exposed in a short space okay I would see and his father is famous because he is an author of inflation this inflation of the universe G-U-T-H this is famous Goose famous name this inflation of the universe his creation and also just on the internet I found he won the the worst possible office largest mess in his office and it's indeed impressive we all good at that but he is and this really impressive it was like that in the level of books in the book average in the room before I saw Milner Milner was pretty good at that also but nothing compared to Goose and a lot easier you can find his articles and I give references probably in my text to his papers Lighty paper is something those seven I guess oh he done Lighty made papers on many different pills but young man but he is quite broad and his first work was I think on actually geometry and then he was doing that it's also a mini mark something mysterious powers but if you look at his name find his papers on the net and I'm saying I'm referring to his paper I'm sure he and but he is not concerned with packing he is concerned with cycles and packing in my view have more much more just to them and of course there are other comparable problems so my subject which I didn't touch basically because my understanding what I can say is will be kind of extremely vague is what is probability 30 for less so he is huge symmetry and this is just you have more symmetry than than in classical probabilities topology has more symmetries and the results you extend them they become more powerful and more general due to the symmetry on the other hand in the application the major application will be languages and so what is probability in languages what is probability of a sentence and that's I don't know answer I was thinking about that and I have response also elaborate what is probability of a sentence and you can't give an answer but you will be not a number you have to create both create two kind category type of objects one related to languages and one related to domain where this range is probability to take and then to say something and it's quite interesting so I can give an example which I like very much so because you can make some experiments on Google and they see how plausible this and of course you cannot say exactly what probability is but if one appears 10 times or 20 times more often well it tells you something and this is a kind of very typical so one is some cats eat mice many cats eat grass so what is will be more common this one by far more common and these are most non-existent this is about you cannot trust mathematics mathematicians say about first is true, second is questionable first is so trivial you don't say it and the second is not trivial therefore you say it often actually most cats is grass probably I forgot each one of them and this will be 10 times more frequent and this will be close to 0 of course Google but be careful the number it gives it's unclear what you mean and then so the range is very tricky even on the day when it comes to numbers of course numbers are on the light in a very rough way numbers are there like ranks of groups homology groups or more elementary example if you look at this too peaks fly or bats fly and this is much more because this on the internet will be more common we have more flying peaks but in bats if you naively look at the test but then exactly there you see the counting is wrong you cannot count by numbers because if you count correctly this will be infinitely more peaks don't fly you can deduce it from what they say the internet peaks don't fly because on one hand it's said many times but if you look at it correctly and this will probability it will be roughly speaking here this will be combinatorics like that for bats and for peaks it will be like that with one point it will be thick point but it will be just a point usually appeared in the same sentences always the same very few constructions the number of grammatical constructions you never saw yesterday I saw a peak flying in my window don't say but for bats you have the number of variations of constructions the number of constructions for peaks it will be negligible for bats it will be as many as for birds their frequency how much they use in material what construction, what combinatorics they are being involved into and this was a kind of now people kind of realize that but relatively recently when the first corpore of legwies were made the word thou if I write correctly or thou, right? or you write this way it was a kind of force most common word in English because in certain sense oh yeah, that would be very common I mean but again it will be always the same construction it's not the number, you see there how many particular book was published doesn't make the word there more common and this applies to many things related to learning also when you look at so called bison approach to statistic people say all these words but they kind of still try to express it's not true learning mechanism of course is bison strategic meaning but it's not moving in numbers much more elaborate structure and for each particular instance you have particular class of problems you have to guess formalize structure I thought a little bit for languages but it's not I see examples I can't explain it but I don't see how to formalize it properly formalization when you formalize it, there is some problem with that and it should be treated differently but this unfortunately I don't know much to say I can't talk about this infinite amount of time without saying anything that's it's unclear to me how to properly express it but when it comes to pure mathematics there are many avenues of very changing probability diverting from this standard standard way of doing it which is makes it quite interesting but again I keep talking and I expect your questions of course you know there is a small part at the end of large variations this small part on large deviations yes about large deviations ok, so large deviations are very general terms classical large deviation is when you have a family of probability distributions on given space X so for example when you have probability given by some densities and and then you want to know what the limit is and typically they for example may be associated to the sum of independent random variables so they become kind of concentrated sharper and sharper in the exponentially decay and the question is how to determine this rate of decay and the first result of that was due to well it's usually people refer to Boltzmann some refer to Varadana and some refer to Euler essentially asymptotic binomial coefficients when you have independent random variable and so in fact for general if you have just measure so it convolves measure with itself and you properly scale it and take log and divide by n and what happens then you know that you have this then you know that this limit that so so I keep confusing it either determines its Ligand transform can be determined by Laplace asymptotic of Laplace transform of this sequence right so if you can see Laplace transform of the sequence properly normalized then not surprisingly it's kind of standard thing you get Ligand transform of this limit this is one of the basic formulas probably known to Gibbs I know I learned it in Texas but certainly written by Lanford so this one of the main results and also elaboration of that and I don't quite understand kind of what is that in what sense it's different from that on one hand on the other hand in geometry what you face when space also is very and when space also variable it's much more subtle so what you look at so here as I said you may consider not independent variable but weakly dependent variable on statistical mechanics for which you want to know what behavior is for the general principle essentially much of mathematical statistical mechanics is about large deviation because this is entropy after all right which I look at Browth through only superficiality but this kind of a major source he is I say the clearest right or what I sorry about that I think Alice maybe confusion maybe not Alice I think I am giving the wrong probably the wrong name maybe Alice here are some books called large deviation statistical mechanics and if you look at large deviation on the internet you come across his article and his book but in geometry so which is we have for example even kind of very simple cases and you can see the dimension dimension going and you want to understand how measure distributes is one and so maybe concentrated somewhere but somewhere exponentially decay but the full picture is not that clear so where measure is sitting how it depends how you look at it here you already have a function so the function is observable which you have and here this observable must be introduced and the observable have natural function you have here like distance to the center and all of that I don't know but I believe there is a good language saying what the limit is and mathematical of this language is will be of course this tropical geometry you come from you see an exponential rate growing on decay of something well algebraically organized so they are well algebraically organized it will be tropical if not it will be something more less elegant maybe not so preachy there are lots of instances beside this sequence we look sequence of compact symmetric spaces of a particular type and you see how also the geometry distributed so your basic function coming from distances or coming from eigenfunctions how they distributed what are the asymptotics and this has been hardly attached this has been studied of course quite a bit of simplices in biology because this is a space of distribution of different phenotypes, genotypes the sphere we understand pretty well sphere in many respects because it is essential for local space geometry general symmetric space I don't think they had been studied in a proper way except what follows trivially from what we know about sphere or something similar there was a recent very beautiful result of that kind concerning this polyhedron of bistochastic matrices in the space of bistochastic matrices how this may be distributed I think there was a talk by Talagran on spin glasses because there was a conjecture by Fees if I got exactly what it says exactly of this nature how measure is distributed on this polyhedron in terms of particular observables and there was explicit formula conjectured by them and was proven by Talagran but some of them addition so it's not another very interesting space which is not at all situation is not not at all clear but of course I'm quite ignorant of that I know a little bit about this geometric examples but I don't know the field of the whole field because at the beginning what they do it's elaborating this Boyceman formula and they emphasize in my view what is the formulas for these people are usually obvious and then some analysis they can foundational they can see the most subtle for me is opposite this what they call subtle for me is obvious so their formula is looking very complicated so for me hard to appreciate what they do at the beginning and then they come go go go and I never followed it so I cannot tell much but there is a now good book from school on algebraic statistics when he systematically exploit some of these aspect of that related to topic of geometry statistical thing because many classical function like better function gamma function they all have this origin they exactly distribution of particular measures on particular shapes and this are this kind of polyhidren but then there are these quantum object like quantum meaning концы, как концы и концы в космосе. Я дам некоторые референсы, я думаю, что это книга Барабиско Барабиско, его имя, который написал неправильный артик про эту информацию о геометрии и классическом демоне. Это также связано с геометрией этой фишеметрии в каких-то instances. Но, опять же, мой знаменитый, довольно суперфишер, я знаю немного, но я не знаю этого, да. Так, кажется, это может быть больший сочет, да, в различные направлениях в этом, который долго прихистое и движет.